This is something I've been wanting to post for a while! This is how I sew my FOE into a continuous loop. I've posted a video on my YouTube channel, but I know it can be hard to see exactly what I am doing... so, I hope this is helpful!
First, I cut the end of my elastic so that it overlaps my start-point by about 1/2" (keeping the same amount of tension on the elastic as you have done over the rest of your opening). I fold the raw end under by about 1/4".
Stitch right to the edge of the fold and lower your needle into the elastic. Raise the foot and pivot your garment so that the you can now zig-zag stitch over your folded edge, creating a nicely finished join. Backtack at the end of your stitching.
After that, just trim back any excess elastic on the wrong side and away you go! This area tends to get bulky so you may need to help it through your machine a little - I know my machine sometimes gets a little caught up on the bump that the join creates.
I hope this is helpful! Pretty much all of my patterns use Fold Over Elastic for edging. I like to use it because it comes in a million different colours, so it's easy to match with almost any fabric, but it can be tricky to finish.